It is shown that all polar encoding schemes of rate $R>\frac{1}{2}$ of block length $N$ implemented according to the Thompson VLSI model must take energy $E\geΩ\left(N^{3/2}\right)$. This lower bound is achievable up to polylogarithmic factors using a mesh network topology defined by Thompson and the encoding algorithm defined by Arikan. A general class of circuits that compute successive cancellation decoding adapted from Arikan's butterfly network algorithm is defined. It is shown that such decoders implemented on a rectangle grid for codes of rate $R>2/3$ must take energy $E\geΩ(N^{3/2})$, and this can also be reached up to polylogarithmic factors using a mesh network. Capacity approaching sequences of energy optimal polar encoders and decoders, as a function of reciprocal gap to capacity $χ= (1-R/C)^{-1}$, have energy that scales as $Ω\left(χ^{5.325}\right)\le E \le O\left(χ^{7.05}\log^{4}\left(χ\right)\right)$.
Fixed-complexity Sphere Decoder (FSD) is a recently proposed technique for Multiple-Input Multiple-Output (MIMO) detection. It has several outstanding features such as constant throughput and large potential parallelism, which makes it suitable for efficient VLSI implementation. However, to our best knowledge, no VLSI implementation of FSD has been reported in the literature, although some FPGA prototypes of FSD with pipeline architecture have been developed. These solutions achieve very high throughput but at very high cost of hardware resources, making them impractical in real applications. …
Thompson's model of VLSI computation relates the energy of a computation to the product of the circuit area and the number of clock cycles needed to carry out the computation. It is shown that for any family of circuits implemented according to this model, using any algorithm that performs decoding of a codeword passed through a binary erasure channel, as the block length approaches infinity either (a) the probability of block error is asymptotically lower bounded by 1/2 or (b) the energy of the computation scales at least as Omega(n(log n)^(1/2)), and so the energy of successful decoding, per…
This paper presents a new VLSI friendly framework for scalable video coding based on Compressed Sensing (CS). It achieves scalability through 3-Dimensional Discrete Wavelet Transform (3-D DWT) and better compression ratio by exploiting the inherent sparsity of the high-frequency wavelet sub-bands through CS. By using 3-D DWT and a proposed adaptive measurement scheme called AMS at the encoder, one can succeed in improving the compression ratio and reducing the complexity of the decoder. The proposed video codec uses only 7% of the total number of multipliers needed in a conventional CS-based v…
Motivated by recently derived fundamental limits on total (transmit + decoding) power for coded communication with VLSI decoders, this paper investigates the scaling behavior of the minimum total power needed to communicate over AWGN channels as the target bit-error-probability tends to zero. We focus on regular-LDPC codes and iterative message-passing decoders. We analyze scaling behavior under two VLSI complexity models of decoding. One model abstracts power consumed in processing elements ("node model"), and another abstracts power consumed in wires which connect the processing elements ("w…
Successive cancellation list (SCL) decoding algorithm is a powerful method that can help polar codes achieve excellent error-correcting performance. However, the current SCL algorithm and decoders are based on likelihood or log-likelihood forms, which render high hardware complexity. In this paper, we propose a log-likelihood-ratio (LLR)-based SCL (LLR-SCL) decoding algorithm, which only needs half the computation and storage complexity than the conventional one. Then, based on the proposed algorithm, we develop low-complexity VLSI architectures for LLR-SCL decoders. Analysis results show that…
Ultra-reliable low-latency communication (URLLC), a major 5G New-Radio use case, is the key enabler for applications with strict reliability and latency requirements. These applications necessitate the use of short-length and high-rate codes. Guessing Random Additive Noise Decoding (GRAND) is a recently proposed Maximum Likelihood (ML) decoding technique for these short-length and high-rate codes. Rather than decoding the received vector, GRAND tries to infer the noise that corrupted the transmitted codeword during transmission through the communication channel. As a result, GRAND can decode a…
In this letter, we propose parallel SC (Successive Cancellation) decoder and parallel SC-List decoder for polar codes. The parallel decoder is composed of M=2^m(m>=1) component decoders working in parallel and each component decoder decodes a Polar code of a block size of 1/M of the original Polar code. Therefore the parallel decoder has M times faster decoding speed. Our simulation results show that the parallel decoder has almost the same error-rate performance as the conventional non-parallel decoder.
Polar codes, as the first provable capacity-achieving error-correcting codes, have received much attention in recent years. However, the decoding performance of polar codes with traditional successive-cancellation (SC) algorithm cannot match that of the low-density parity-check (LDPC) or turbo codes. Because SC list (SCL) decoding algorithm can significantly improve the error-correcting performance of polar codes, design of SCL decoders is important for polar codes to be deployed in practical applications. However, because the prior latency reduction approaches for SC decoders are not applicab…
We propose a nonuniform quantized decoder for polar codes. The design metric of the quantizers is to minimize the distortion incurred by quantization. The quantizers are obtained via dynamic programming and the optimality of the quantizer is proved as well. Simulation results show that the error correction performance degradation of the proposed nonuniform quantized decoder is less than 0.1 dB compared to conventional float-point decoders under 5-bit nonuniform quantization, outperforming previous uniform quantized decoders for polar codes significantly.
Polar codes has been selected as 5G standard. However, only a couple of ASIC featuring decoders are fabricated,and none of them support list size L > 4 and code length N > 1024. This paper presents an ASIC implementation of three decoders for polar code: successive cancellation (SC) decoder, flexible decoder and ultra-reliable decoder. These decoders are all SC based decoder, supporting list size up to 1,8,32 and code length up to 2^15,2^14,2^11 respectively. This chip is fabricated in a 16nm TSMC FinFET technology, and can be clocked at 1 Ghz. Optimization techniques are proposed and employed…
In this letter, we propose an adaptive SC (Successive Cancellation)-List decoder for polar codes with CRC. This adaptive SC-List decoder iteratively increases the list size until the decoder outputs contain at least one survival path which can pass CRC. Simulation shows that the adaptive SC-List decoder provides significant complexity reduction. We also demonstrate that polar code (2048, 1024) with 24-bit CRC decoded by our proposed adaptive SC-List decoder with very large list size can achieve a frame error rate FER=0.001 at Eb/No=1.1dB, which is about 0.2dB from the information theoretic lim…
Polar codes have promising error-correction capabilities. Yet, decoding polar codes is often challenging, particularly with large blocks, with recently proposed decoders based on list-decoding or neural-decoding. The former applies multiple decoders or the same decoder multiple times with some redundancy, while the latter family utilizes emerging deep learning schemes to learn to decode from data. In this work we introduce a novel polar decoder that combines the list-decoding with neural-decoding, by forming an ensemble of multiple weighted belief-propagation (WBP) decoders, each trained to de…
Arıkan's polar coding, is by now a well studied technique that allows achieving the symmetric capacity of binary input memoryless channels with low complexity encoding and decoding, provided that the polar decoding architecture is used and the decoding metric is matched to the true channel. In this paper, we analyze communication rates that are achievable when the polar coding/decoding architecture is used with the decoder using an incorrect model of the channel. We define the `polar mismatched capacity' as an analogue of the classical mismatched capacity, give an expression for it, and derive…
Polar codes are of great interest since they are the first provably capacity-achieving forward error correction codes. To improve throughput and to reduce decoding latency of polar decoders, maximum likelihood (ML) decoding units are used by successive cancellation list (SCL) decoders as well as successive cancellation (SC) decoders. This paper proposes an approximate ML (AML) decoding unit for SCL decoders first. In particular, we investigate the distribution of frozen bits of polar codes designed for both the binary erasure and additive white Gaussian noise channels, and take advantage of th…
Successive cancellation list decoders with flip operations (SCL-Flip) can utilize re-decoding attempts to significantly improve the error-correction performance of polar codes. However, these re-decoding attempts result in extra computation complexity, which thus leads to increased energy consumption and decoding latency to the communication system adopting SCL-Flip decoders. To significantly reduce the computation complexity of current SCL-Flip decoders, we design a new adaptive SCL-Flip (AD-SCLF) decoder, which can be easily implemented based on existing SCL-Flip techniques. Simulation resul…
Under successive cancellation (SC) decoding, polar codes are inferior to other codes of similar blocklength in terms of frame error rate. While more sophisticated decoding algorithms such as list- or stack-decoding partially mitigate this performance loss, they suffer from an increase in complexity. In this paper, we describe a new flavor of the SC decoder, called the SC flip decoder. Our algorithm preserves the low memory requirements of the basic SC decoder and adjusts the required decoding effort to the signal quality. In the waterfall region, its average computational complexity is almost …
In this paper, we propose a decision-aided scheme for parallel SC-List decoding of polar codes. At the parallel SC-List decoder, each survival path is extended based on multiple information bits, therefore the number of split paths becomes very large and the sorting to find the top L paths becomes very complex. We propose a decision-aided scheme to reduce the number of split paths and thus reduce the sorting complexity.
Due to their capacity-achieving property, polar codes have become one of the most attractive channel codes. To date, the successive cancellation list (SCL) decoding algorithm is the primary approach that can guarantee outstanding error-correcting performance of polar codes. However, the hardware designs of the original SCL decoder have large silicon area and long decoding latency. Although some recent efforts can reduce either the area or latency of SCL decoders, these two metrics still cannot be optimized at the same time. This paper, for the first time, proposes a general log-likelihood-rati…
Polar codes, discovered by Arıkan, are the first error-correcting codes with an explicit construction to provably achieve channel capacity, asymptotically. However, their error-correction performance at finite lengths tends to be lower than existing capacity-approaching schemes. Using the successive-cancellation algorithm, polar decoders can be designed for very long codes, with low hardware complexity, leveraging the regular structure of such codes. We present an architecture and an implementation of a scalable hardware decoder based on this algorithm. This design is shown to scale to code le…
Multi-kernel polar codes have recently been proposed to construct polar codes of lengths different from powers of two. Decoder implementations for multi-kernel polar codes need to account for this feature, that becomes critical in memory management. We propose an efficient, generalized memory management framework for implementation of successivecancellation decoding of multi-kernel polar codes. It can be used on many types of hardware architectures and different flavors of SC decoding algorithms. We illustrate the proposed solution for small kernel sizes, and give complexity estimates for vari…
This paper presents an efficient hardware design approach for list successive cancellation (LSC) decoding of polar codes. By applying path-overlapping scheme, the l instances of (l > 1) successive cancellation (SC) decoder for LSC with list size l can be cut down to only one. This results in a dramatic reduction of the hardware complexity without any decoding performance loss. We also develop novel approaches to reduce the latencyassociated with the pipeline scheme. Simulation results show that with proposed design approach the hardware efficiency is increased significantly over the recently p…
Previous work showed that polar codes can be decoded using off-the-shelf LDPC decoders by imposing special constraints on the LDPC code structure, which, however, resulted in some performance degradation. In this paper we show that this loss can be mitigated; in particular, we demonstrate how the gap between LDPC-style decoding and Arikan's Belief Propagation (BP) decoding of polar codes can be closed by taking into account the underlying graph structure of the LDPC decoder while jointly designing the polar code and the parity-check matrix of the corresponding LDPC-like code. The resulting pol…
Polar codes have gained extensive attention during the past few years and recently they have been selected for the next generation of wireless communications standards (5G). Successive-cancellation-based (SC-based) decoders, such as SC list (SCL) and SC flip (SCF), provide a reasonable error performance for polar codes at the cost of low decoding speed. Fast SC-based decoders, such as Fast-SSC, Fast-SSCL, and Fast-SSCF, identify the special constituent codes in a polar code graph off-line, produce a list of operations, store the list in memory, and feed the list to the decoder to decode the co…
This paper focuses on the recently introduced Successive Cancellation Flip (SCFlip) decoder of polar codes. Our contribution is twofold. First, we propose the use of an optimized metric to determine the flipping positions within the SCFlip decoder, which improves its ability to find the first error that occurred during the initial SC decoding attempt. We also show that the proposed metric allows closely approaching the performance of an ideal SCFlip decoder. Second, we introduce a generalisation of the SCFlip decoder to a number of $ω$ nested flips, denoted by SCFlip-$ω$, using a similar optim…
We propose a belief propagation list (BPL) decoder with comparable performance to the successive cancellation list (SCL) decoder of polar codes, which already achieves the maximum likelihood (ML) bound of polar codes for sufficiently large list size $L$. The proposed decoder is composed of multiple parallel independent belief propagation (BP) decoders based on differently permuted polar code factor graphs. A list of possible transmitted codewords is generated and the one closest to the received vector, in terms of Euclidean distance, is picked. To the best of our knowledge, the proposed BPL de…
This paper presents a hardware architecture of fast simplified successive cancellation (fast-SSC) algorithm for polar codes, which significantly reduces the decoding latency and dramatically increases the throughput. Algorithmically, fast-SSC algorithm suffers from the fact that its decoder scheduling and the consequent architecture depends on the code rate; this is a challenge for rate-compatible system. However, by exploiting the homogeneousness between the decoding processes of fast constituent polar codes and regular polar codes, the presented design is compatible with any rate. The schedu…
Polar codes is promising because they can provably achieve the channel capacity while having an explicit construction method. Lots of work have been done for the bit-based decoding algorithm for polar codes. In this paper, generalized symbol-based successive cancellation (SC) and SC list decoding algorithms are discussed. A symbol-based recursive channel combination relationship is proposed to calculate the symbol-based channel transition probability. This proposed method needs less additions than the maximum-likelihood decoder used by the existing symbol-based polar decoding algorithm. In add…
Polar codes are of great interests because they provably achieve the capacity of both discrete and continuous memoryless channels while having an explicit construction. Most existing decoding algorithms of polar codes are based on bit-wise hard or soft decisions. In this paper, we propose symbol-decision successive cancellation (SC) and successive cancellation list (SCL) decoders for polar codes, which use symbol-wise hard or soft decisions for higher throughput or better error performance. First, we propose to use a recursive channel combination to calculate symbol-wise channel transition pro…
Polar codes are the first provably capacity-achieving forward error correction codes. To improve decoder throughput, the symbol-decision SC algorithm makes hard-decision for multiple bits at a time. In this paper, we prove that for polar codes, the symbol-decision SC algorithm is better than the bit-decision SC algorithm in terms of the frame error rate (FER) performance because the symbol-decision SC algorithm performs a local maximum likelihood decoding within a symbol. Moreover, the bigger the symbol size, the better the FER performance. Finally, simulation results over both the additive wh…
Polar codes have emerged as the most favorable channel codes for their unique capacity-achieving property. To date, numerous works have been reported for efficient design of polar codes decoder. However, these prior efforts focused on design of polar decoders via deterministic computation, while the behavior of stochastic polar decoder, which can have potential advantages such as low complexity and strong error-resilience, has not been studied in existing literatures. This paper, for the first time, investigates polar decoding using stochastic logic. Specifically, the commonly-used successive …
Flexibility is one mandatory aspect of channel coding in modern wireless communication systems. Among other things, the channel decoder has to support several code lengths and code rates. This need for flexibility applies to polar codes that are considered for control channels in the future 5G standard. This paper presents a new generic and flexible implementation of a software Successive Cancellation List (SCL) decoder. A large set of parameters can be fine-tuned dynamically without re-compiling the software source code: the code length, the code rate, the frozen bits set, the puncturing patt…
Even though polar codes were adopted in the latest 5G cellular standard, they still have the fundamental problem of high decoding latency. Aiming at solving this problem, a fast simplified successive cancellation (Fast-SSC) decoder based on the new class of sequence repetition (SR) nodes has been proposed recently in \cite{sr2020} and has a lower required number of time steps than other existing Fast-SSC decoders in theory. This paper focuses on the hardware implementation of this SR node-based fast-SSC (SRFSC) decoder. The implementation results for a polar code with length 1024 and code rate…
This work is on fast encoding and decoding of polar codes. We propose and detail 8-bit and 16-bit parallel decoders that can be used to reduce the decoding latency of the successive-cancellation decoder. These decoders are universal and can decode flexible-rate and flexible-length polar codes. We also present fast encoders that can be used to increase the throughput of serially-implemented polar encoders.
Polar codes are newly discovered capacity-achieving codes, which have attracted lots of research efforts. Polar codes can be efficiently decoded by the low-complexity successive cancelation (SC) algorithm and the SC list (SCL) decoding algorithm. The belief propagation (BP) decoding algorithm not only is an alternative to the SC and SCL decoders, but also provides soft outputs that are necessary for joint detection and decoding. Both the BP decoder and the soft cancelation (SCAN) decoder were proposed for polar codes to output soft information about the coded bits. In this paper, first a belie…
As a new joint source-channel coding scheme, the double polar (D-Polar) codes have been proposed recently. In this letter, a novel joint source-channel decoder, namely the joint successive cancellation list (J-SCL) decoder, is proposed to improve the decoding performance of the D-Polar codes. We merge the trellis of the source polar code and that of the channel polar code to construct a compound trellis. In this compound trellis, the joint source-channel nodes represent both of the information bits and the high-entropy bits. Based on the compound trellis, the J-SCL decoder is designed to recov…
We show that the performance of iterative belief propagation (BP) decoding of polar codes can be enhanced by decoding over different carefully chosen factor graph realizations. With a genie-aided stopping condition, it can achieve the successive cancellation list (SCL) decoding performance which has already been shown to achieve the maximum likelihood (ML) bound provided that the list size is sufficiently large. The proposed decoder is based on different realizations of the polar code factor graph with randomly permuted stages during decoding. Additionally, a different way of visualizing the p…
A successive cancellation list (SCL) decoder with limited list size for polar codes can not be analyzed as a successive cancellation (SC) decoder, nor as a maximum likelihood (ML) decoder, due to the complicated decoding errors caused by path elimination. To address this issue, an analytical tool, named as cluster pairwise error probability (CPEP), is proposed in this paper to measure the competitiveness of the correct path against the error paths in an SCL decoder. It is shown that the sum of CPEPs over error paths could be used as an indicator of the probability of correct path being elimina…
Successive cancellation (SC) is the first and widely known decoder of polar codes, which has received a lot of attentions recently. However, its decoding schedule generating algorithms are still primitive, which are not only complex but also offline. This paper proposes a simple and online algorithm to generate the decoding schedule of SC decoder. Firstly, the dependencies among likelihood ratios (LR) are explored, which lead to the discovery of a sharing factor. Secondly, based on the online calculation of the sharing factor, the proposed algorithm is presented, which is neither based on the …
A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a $l$-expressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities $W_{G}^{(\cdot)}(\cdot|0)$ and $W_{G}^{(\cdot)}(\cdot|1)$ separately, a $W$-expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a $m\…
Polar codes are a family of capacity-achieving error-correcting codes, and they have been selected as part of the next generation wireless communication standard. Each polar code bit-channel is assigned a reliability value, used to determine which bits transmit information and which parity. Relative reliabilities need to be known by both encoders and decoders: in case of multi-mode systems, where multiple code lengths and code rates are supported, the storage of relative reliabilities can lead to high implementation complexity. In this work, observe patterns among code reliabilities. We propos…
Polar codes provably achieve the capacity of a wide array of channels under successive decoding. This assumes infinite precision arithmetic. Given the successive nature of the decoding algorithm, one might worry about the sensitivity of the performance to the precision of the computation. We show that even very coarsely quantized decoding algorithms lead to excellent performance. More concretely, we show that under successive decoding with an alphabet of cardinality only three, the decoder still has a threshold and this threshold is a sizable fraction of capacity. More generally, we show that …
Polar codes under successive cancellation decoding proposed by Arıkan provably achieve the symmetric capacity of any given binary-input discrete memoryless channel. The successive cancellation list decoder for polar codes was described by Tal and Vardy as a generalization of the successive cancellation decoder of Arıkan. The performance of the successive cancellation list decoder is encouraging in practice. In this paper, we formalize the successive cancellation list decoder in our notation and prove that polar codes under successive cancellation list decoding achieve the symmetric capacity of…
Belief propagation (BP) is an iterative decoding algorithm for polar codes which can be parallelized effectively to achieve higher throughput. However, because of the presence of error floor due to cycles and stopping sets in the factor graph, the performance of the BP decoder is far from the performance of state of the art cyclic redundancy check (CRC) aided successive cancellation list (CA-SCL) decoders. It has been shown that successive BP decoding on multiple permuted factor graphs, which is called the multi-trellis BP decoder, can improve the error performance. However, when permuting the…
In the standard polar code construction, the message vector $(U_0,U_1,\dots,U_{n-1})$ is divided into information bits and frozen bits according to the reliability of each $U_i$ given $(U_0,U_1,\dots,U_{i-1})$ and all the channel outputs. While this reliability function is the most suitable measure to choose information bits under the Successive Cancellation (SC) decoder, there is a mismatch between this reliability function and the Successive Cancellation List (SCL) decoder because the SCL decoder also makes use of the information from the future frozen bits. We propose a Dynamic Programming …
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional w…
Polar codes provably achieve the symmetric capacity of a memoryless channel while having an explicit construction. This work aims to increase the throughput of polar decoder hardware by an order of magnitude relative to the state of the art successive-cancellation decoder. We present an algorithm, architecture, and FPGA implementation of a gigabit-per-second polar decoder.
